Math Problem Statement
Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.)
(x^4-3+6/x^7)dx
Solution
To evaluate the integral of , we'll integrate each term separately.
Step 1: Set up the integral
= \int x^4 \, dx - \int 3 \, dx + \int \frac{6}{x^7} \, dx$$ ### Step 2: Integrate each term 1. **First term:** $$\int x^4 \, dx$$ - Using the power rule $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, where $$n \neq -1$$: $$\int x^4 \, dx = \frac{x^{4+1}}{4+1} = \frac{x^5}{5}$$ 2. **Second term:** $$\int -3 \, dx$$ - Since $$-3$$ is a constant, the integral is: $$\int -3 \, dx = -3x$$ 3. **Third term:** $$\int \frac{6}{x^7} \, dx$$ - Rewrite $$\frac{6}{x^7}$$ as $$6x^{-7}$$ and then apply the power rule: $$\int 6x^{-7} \, dx = 6 \cdot \frac{x^{-7+1}}{-7+1} = 6 \cdot \frac{x^{-6}}{-6} = -x^{-6} = -\frac{1}{x^6}$$ ### Step 3: Combine the results Putting it all together, we get: $$\int \left(x^4 - 3 + \frac{6}{x^7}\right) \, dx = \frac{x^5}{5} - 3x - \frac{1}{x^6} + C$$ ### Step 4: Check by Differentiating To verify, we differentiate the result: $$\frac{d}{dx} \left( \frac{x^5}{5} - 3x - \frac{1}{x^6} + C \right)$$ 1. **Differentiating** $$\frac{x^5}{5}$$: $$\frac{d}{dx}\left(\frac{x^5}{5}\right) = x^4$$ 2. **Differentiating** $$-3x$$: $$\frac{d}{dx}(-3x) = -3$$ 3. **Differentiating** $$-\frac{1}{x^6}$$: Rewrite as $$-x^{-6}$$: $$\frac{d}{dx}(-x^{-6}) = 6x^{-7} = \frac{6}{x^7}$$ Combining these, we recover the original function: $$x^4 - 3 + \frac{6}{x^7}$$ Thus, the integral is correct: $$\int \left(x^4 - 3 + \frac{6}{x^7}\right) \, dx = \frac{x^5}{5} - 3x - \frac{1}{x^6} + C$$ Would you like further details on any step, or do you have any questions? ### Related Questions 1. How would you evaluate $$\int (2x^3 - \frac{5}{x^2}) \, dx$$? 2. Can you verify the integral of $$\int (x^2 + 4x + 6) \, dx$$ by differentiating? 3. What happens if you integrate $$\int (4x^5 - 8 + \frac{10}{x^3}) \, dx$$? 4. How do you evaluate $$\int (x^4 - \frac{3}{x^2} + 7) \, dx$$? 5. How would you handle a more complex integral like $$\int (x^7 + x^{-4} - 5x + 3) \, dx$$? **Tip:** Checking an integral by differentiating your answer is a reliable method to confirm correctness.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Integration
Power Rule
Differentiation
Constant of Integration
Formulas
Integral of x^n = x^(n+1) / (n+1) + C
Power rule for differentiation: d/dx[x^n] = n * x^(n-1)
Theorems
Power Rule for Integration
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12